I don't think cPanel allows access to the backups other than to be able to restore them.
For mysql, you could setup a cron to create a backup that you could download.
Code:
mysqldump --opt -u <cPanel user> -p <cPanel pass> <database name> >backup.sql
So if your cPanel login was
andychev
$3cr37
and your database was
main (which in reality is called andychev_main)
You would do this:
Code:
mysqldump --opt -u andychev -p $3cr37 andychev_main>/home/andychev/www/backup.sql
You might want to consider creating a mysql user with read access to the database and use that instead of having your cpanel info sitting in a cron file, though its probably not a real big deal.